Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
95 views

Proof that the Rubik’s Cube group is 2-generated

Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube: Frank Barnes observes that the group of the cube is generated by two moves: \begin{align*} \alpha &= L^2 B R D^{-1} ...
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3answers
941 views

Random solving of a Rubik cube .

After playing a little with a Rubik cube I thought of the following problem : Suppose we start with a solved Rubik cube (a general one , with $n^3$ cubes) . Then we choose one of the moves , each ...
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1answer
72 views

Solving a Rubik's cube in 20 moves or less

Most "cubers" solve a Rubik's cube in about 60 moves, using special algorithms and quick fingers. But any Rubik's cube can be solved in 20 moves or less (cube20.org). Is it possible that there is a ...
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1answer
116 views

What are some Group representation of the rubik's cube group?

The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)? ...
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2answers
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Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back ...
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1answer
24 views

Can I place the rubiks3cube pieces in the distorted position I intend to get?

Right now I am trying to get the distorted position like this: in each face only one diagonal is solved and no similar colour is on a face other than the diagonal pieces mentioned previously. For ...
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39 views

Rubik's Cube with a missing color

Suppose a color of the Rubik's cube(3x3) is missing, is it possible to find the missing color? Now a brute force method would be solve for all other colors, but that is quite naive.
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When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
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Probability function for centerpieces of a 4×4 Rubik's cube

I have a question regarding blindsolving of a 4×4 Rubik's cube (a.k.a 4-blind). When a 4-blind solve begins, one is free to pick any rotation of the 4×4 cube. To minize effort needed to solve the ...
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1answer
159 views

Asymptotic value of permutations of general Rubik Cube

I found this on C|NET, and wondering if there was anything like a $1000\times1000\times1000$ cube? And how many arrangements would it have? Or if this is too much, how about an asymptotic formula for ...
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1answer
38 views

Homomorphism from Rubik's Cube group to Symmetric Group.

$R^2$ is the rotation of the right side by 180, $F^2$ is the rotation of the front by 180. Other than this I am completely lost and this blowing my mind. I wish I had more to build off of, but I am ...
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1answer
73 views

The mathematics underlying Rubik's games

I am interested in knowing (a little more) about the mathematics underlying some of Erno Rubik's games. I guess we all know his famous cube. I heard at some point that a solution was possible because ...
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2answers
65 views

Distance between two positions of the rubik cube .

It's now known that to reach the solved position of the rubik cube from any other position you need at most $20$ moves (a $180^{\circ}$ of a face is counted as one move , not two ) . See for example ...
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1answer
108 views

Solving a Rubik's Cube Using Calculus or differentiation? [closed]

I have this question to solve a Rubik's cube using calculus or differentiation and then figuring out whats the shortest time it could be solved in...please help me???
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1answer
120 views

What do I need to learn to solve the math behind a Rubiks cube?

I stopped studying math at precalc. What should I study to be able to understand how to solve a Rubics cube using mathematics? I don't think this is an opinion question, and I really want to learn ...
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2answers
227 views

How many ways are there to solve a Rubiks cube? [closed]

Friends have been arguing that a rubiks cube could never be mastered cause there are too many different ways to solve it cause there are 43,252,003,274,489,856,000 positions the Cube could have, so ...
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1answer
151 views

How many combinations in 10x10x10 Rubik's cube?

I was wondering how many possible combinations there is in the cubes greater than 3x3x3 (4x4x4, 5x5x5, ..., 10x10x10)? We know that in 3x3x3 there are about 4,3 * 10^19 combinations, what about bigger ...
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2answers
85 views

Rubik's Cube's Group

Is there an article somewhere with an exhaustive study of the Rubik's Cube Group $G$? Such as computing some subgroups of it or exhibiting some elements of its center $Z(G)$? I tried googling it and ...
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1answer
108 views

Rubik's Cube Thistlethwaite Four Phase Algorithm

I am trying to understand The Four-Phase-Algorithm ( Thistlethwaite's 52-move algorithm ) for Rubik's Cube but no easy understandable explaination is given. It would be helpful if given with a small ...
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1answer
34 views

how to evaluate combinations of different configurarations of a rubik's cube

how to calculate the total combination number of an rubik's cube ? and what is the number of combination if we fix one face of it with all matched colours , specifically saying blue one ,?? no. of ...
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1answer
136 views

$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?

I've been working on a problem related to the 3x3x3 Rubik's Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is ...
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42 views

Can solution to Rubik's cube be seen from the point of view of Markov Decision Process?

Solving Rubik's cube can be thought of as a Planning problem which has : a state space $S$ a set $G \subseteq S$ of goal states (in this case singleton) actions $A(s) \subseteq A$ applicable in ...
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1answer
35 views

Imitating smaller Rubik's cubes with bigger ones.

Let's assume we have a 4x4x4 Rubik's cube. With this cube we can imitate a 2x2x2 by considering all 2x2x2 corner cubes as 1 block. Similarly, we can solve it like a 3x3x3 by first solving the 2x2 ...
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233 views

Number of moves necessary to solve Rubik's cube by pure chance

Suppose, random moves are made to solve Rubik's cube. A move consists of a $90$-degree-rotation of some side. The starting position is also random. What is $E(X)$, where $X$ is the number of moves ...
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1answer
132 views

How to learn about the Rubik's Cube?

So I just started learning Abstract Algebra and I've recently appreciated an introduction to group theory. Among the interesting avenues of group theory is found in tackling a Rubik's Cube. I've ...
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1answer
57 views

Help with proving property of Rubik's cube.

I am looking at these notes, and am trying to prove the following on page 13: Let $C_1$ and $C_2$ be two different unoriented corner cubies, and let $C'_{1}$ and $C'_{2}$ be two different ...
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1answer
184 views

Solve All Sequence (Rubik's Cube)

Can you prove/disprove that there is a solve-all sequence of moves to complete the Rubik's Cube from any solvable-position? If so, can you explain how long it is? If not, explain why not. Just to be ...
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2answers
237 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
155 views

Where Does F' in Rubik's Cube Group Singmaster Notation Come From?

Basic 90° : F turns the front clockwise 180° : F^2 turns the front clockwise twice -90° : F' turns the front counter-clockwise Why are we calling the -90° rotation F' and not -F? (source: ...
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77 views

Rubik's cube computing algorithm

Is it possible to solve a 3x3 Rubik's cube using a tree similar to a 'mini-max game tree' (listing all possible unique moves, then listing further moves from this and so on), or is the sample space ...
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0answers
145 views

Minimum number of stickers required for 3x3 Rubik's Cube

Lately, the stickers on my V-Cube have been peeling off, and I became curious: what is the minimum number of stickers I would actually need in order to represent each unique configuration? I have but ...
6
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1answer
409 views

Miniature Rubik's Cube

$R$ is a clock-wise twist of a right-hand face and $U$ is a clock-wise twist of the upper face of a $2 \times 2 \times 2$ Rubik's Cube. Find the order of the subgroup generated by $\{R, U\}$
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How to use group theory to solve larrys square iphone app

There's a 2 d version of rubiks cube on apple app store. How can group theory give an algorithm to solve the iPhone app:larry's square.
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2answers
228 views

Is there a proof, that for any Rubik's cube, after solving the first 2 layers, there will always be an even amount of oriented edges?

I've always wondered if there was a proof for this, since the beginners method to solving the last layer depends on this (you either have no edges oriented correctly, an L, a line or a cross, and each ...
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1answer
256 views

Complex automata Rubiks cube question (with picture) help needed

Question 1 A Rubik’s Cube is a puzzle in the shape of a cube. Each face is covered by nine stickers, each of which is coloured with one of six colours: white, red, blue, orange, green, andyellow. An ...
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1answer
325 views

Number of unique permutations of a 3x3x3 cube

Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted. The ...
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52 views

identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
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1answer
2k views

Creating a Rubik's Cube Algorithm

So, I realize this isn't a very math-oriented question, but I thought it fit better here than anywhere else. Anyway, I was simply wondering how people come up with the many Rubik's Cube algorithms, ...
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53 views

Rubik's cube with different neighboring color

How many valid configuration of the 3x3 Rubik's cube are there so that no two adjacent faces are the same color? Thanks.
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3answers
975 views

Minimum number of random moves needed to uniformly scramble a Rubik's cube?

Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves ...
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3answers
153 views

$3 \times 3$ Rubik's cube scrambling question

It was proven that a Rubik's cube needs at most $20$ moves to solve. This implies that any configuration of a Rubik's cube can be reached from an unscrambled Cube in at most $20$ moves. So, say when ...
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1answer
151 views

Eulerian path for Rubik's Cube states

There are a number of discussions online confirming that there exists a Hamiltonian cycle through the states of a Rubik's Cube. Or more precisely, the "quarter-turn metric Cayley graph for the Rubik's ...
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1answer
2k views

Can RUBIK's cube be solved using group theory?

Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
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1answer
71 views

Independently rotating cubelets of a 2 x 2 rubiks cube?

I was reading up on 2 x 2 Rubik's cube. Wikipedia says that Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). I ...
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1answer
109 views

What is “tetrad twist” in relation to Rubik's cubes?

I am trying to write a computer program to solve Rubik's cubes using a version of Thistlethwaite's algorithm. Instead of using lookup tables to perform each phase of the algorithm, I'm using a ...
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3answers
4k views

Can a Rubik's cube be mapped knowing only two sides?

Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a ...
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1answer
196 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
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2answers
107 views

Rubik cube number of alternative solutions

If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that ...
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2answers
907 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...